Related papers: Witnessed k-Distance
This paper introduces a new way to compact a continuous probability distribution $F$ into a set of representative points called support points. These points are obtained by minimizing the energy distance, a statistical potential measure…
Compared with word embedding based on point representation, distribution-based word embedding shows more flexibility in expressing uncertainty and therefore embeds richer semantic information when representing words. The Wasserstein…
Let $X$ be a metric space and $BCl(X)$ the collection of nonempty bounded closed subsets of $X$. We show that Hausdorff distance $d_H$ belongs to a specific family of real-valued distances on $BCl(X)$, each of which can be expressed as the…
Nonparametric curve estimation by kernel methods has attracted widespread interest in theoretical and applied statistics. One area of conflict between theory and application relates to the evaluation of the performance of the estimators.…
In many robotics applications, it is necessary to compute not only the distance between the robot and the environment, but also its derivative - for example, when using control barrier functions. However, since the traditional Euclidean…
Distance metric learning can be viewed as one of the fundamental interests in pattern recognition and machine learning, which plays a pivotal role in the performance of many learning methods. One of the effective methods in learning such a…
The Wasserstein distance is a distance between two probability distributions and has recently gained increasing popularity in statistics and machine learning, owing to its attractive properties. One important approach to extending this…
This document reviews the definition of the kernel distance, providing a gentle introduction tailored to a reader with background in theoretical computer science, but limited exposure to technology more common to machine learning,…
The analysis of animal movement has gained attention recently. New continuous-time models and statistical methods have been developed to estimate some sets related to their movements, such as the home-range and the core-area among others,…
This paper introduces a novel method to estimate distance fields from noisy point clouds using Gaussian Process (GP) regression. Distance fields, or distance functions, gained popularity for applications like point cloud registration,…
This paper considers the problem of regression over distributions, which is becoming increasingly important in machine learning. Existing approaches often ignore the geometry of the probability space or are computationally expensive. To…
Variational Bayesian inference is an important machine-learning tool that finds application from statistics to robotics. The goal is to find an approximate probability density function (PDF) from a chosen family that is in some sense…
The Wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits. To overcome this obstacle, for probability measures supported on finitely many points, we derive…
We give a definition of thickness in $\mathbb{R}^d$ that is useful even for totally disconnected sets, and prove a Gap Lemma type result. We also guarantee an interval of distances in any direction in thick compact sets, relate thick sets…
The surge of research in image segmentation has yielded remarkable performance gains but also exposed a reproducibility crisis. A major contributor is performance evaluation, where both selection and implementation of metrics play critical…
A stochastic algorithm is proposed, finding the set of generalized means associated to a probability measure on a compact Riemannian manifold M and a continuous cost function on the product of M by itself. Generalized means include p-means…
In any setting in which observable properties have a quantitative flavour, it is natural to compare computational objects by way of \emph{metrics} rather than equivalences or partial orders. This holds, in particular, for probabilistic…
This research work aims to explore the distortions in distance in equidistant cylindrical projection. The horizontal bending that occurs in the projection process can be assessed by performing a geometric analysis using Tissot's…
In this paper, we relate the framework of mod-$\phi$ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is that of Fourier analysis in the Wiener algebra, allowing…
The Hausdorff distance, the Gromov-Hausdorff, the Fr\'echet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as $\inf_\rho…